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74-470: Theoretical chemistry unites principles and concepts common to all branches of chemistry. Within the framework of theoretical chemistry, there is a systematization of chemical laws, principles and rules, their refinement and detailing, the construction of a hierarchy. The central place in theoretical chemistry is occupied by the doctrine of the interconnection of the structure and properties of molecular systems. It uses mathematical and physical methods to explain

148-435: A mathematical theory , in the sense that the word "theory" has a different meaning in mathematical terms. R i c = k g {\displaystyle \mathrm {Ric} =kg} The equations for an Einstein manifold , used in general relativity to describe the curvature of spacetime A physical theory involves one or more relationships between various measurable quantities. Archimedes realized that

222-405: A working hypothesis . Theorem In mathematics and formal logic , a theorem is a statement that has been proven , or can be proven. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and

296-404: A certain economy and elegance (compare to mathematical beauty ), a notion sometimes called " Occam's razor " after the 13th-century English philosopher William of Occam (or Ockham), in which the simpler of two theories that describe the same matter just as adequately is preferred (but conceptual simplicity may mean mathematical complexity). They are also more likely to be accepted if they connect

370-400: A completely symbolic form (e.g., as propositions in propositional calculus ), they are often expressed informally in a natural language such as English for better readability. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which

444-467: A completely symbolic form—with the presumption that a formal statement can be derived from the informal one. It is common in mathematics to choose a number of hypotheses within a given language and declare that the theory consists of all statements provable from these hypotheses. These hypotheses form the foundational basis of the theory and are called axioms or postulates. The field of mathematics known as proof theory studies formal languages, axioms and

518-500: A form of protoscience and others are a form of pseudoscience . The falsification of the original theory sometimes leads to reformulation of the theory. "Thought" experiments are situations created in one's mind, asking a question akin to "suppose you are in this situation, assuming such is true, what would follow?". They are usually created to investigate phenomena that are not readily experienced in every-day situations. Famous examples of such thought experiments are Schrödinger's cat ,

592-405: A formal symbolic proof can in principle be constructed. In addition to the better readability, informal arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, one might even be able to substantiate

666-766: A good example. For instance: " phenomenologists " might employ ( semi- ) empirical formulas and heuristics to agree with experimental results, often without deep physical understanding . "Modelers" (also called "model-builders") often appear much like phenomenologists, but try to model speculative theories that have certain desirable features (rather than on experimental data), or apply the techniques of mathematical modeling to physics problems. Some attempt to create approximate theories, called effective theories , because fully developed theories may be regarded as unsolvable or too complicated . Other theorists may try to unify , formalise, reinterpret or generalise extant theories, or create completely new ones altogether. Sometimes

740-412: A layman. In mathematical logic , a formal theory is a set of sentences within a formal language . A sentence is a well-formed formula with no free variables. A sentence that is a member of a theory is one of its theorems, and the theory is the set of its theorems. Usually a theory is understood to be closed under the relation of logical consequence . Some accounts define a theory to be closed under

814-451: A natural number n for which the Mertens function M ( n ) equals or exceeds the square root of n ) is known: all numbers less than 10 have the Mertens property, and the smallest number that does not have this property is only known to be less than the exponential of 1.59 × 10 , which is approximately 10 to the power 4.3 × 10 . Since the number of particles in the universe

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888-493: A new branch of mathematics: infinite, orthogonal series . Modern theoretical physics attempts to unify theories and explain phenomena in further attempts to understand the Universe , from the cosmological to the elementary particle scale. Where experimentation cannot be done, theoretical physics still tries to advance through the use of mathematical models. Mainstream theories (sometimes referred to as central theories ) are

962-420: A semantics for them through interpretation . Although theorems may be uninterpreted sentences, in practice mathematicians are more interested in the meanings of the sentences, i.e. in the propositions they express. What makes formal theorems useful and interesting is that they may be interpreted as true propositions and their derivations may be interpreted as a proof of their truth. A theorem whose interpretation

1036-498: A ship floats by displacing its mass of water, Pythagoras understood the relation between the length of a vibrating string and the musical tone it produces. Other examples include entropy as a measure of the uncertainty regarding the positions and motions of unseen particles and the quantum mechanical idea that ( action and) energy are not continuously variable. Theoretical physics consists of several different approaches. In this regard, theoretical particle physics forms

1110-637: A single counter-example and so establish the impossibility of a proof for the proposition as-stated, and possibly suggest restricted forms of the original proposition that might have feasible proofs. For example, both the Collatz conjecture and the Riemann hypothesis are well-known unsolved problems; they have been extensively studied through empirical checks, but remain unproven. The Collatz conjecture has been verified for start values up to about 2.88 × 10 . The Riemann hypothesis has been verified to hold for

1184-399: A theorem by using a picture as its proof. Because theorems lie at the core of mathematics, they are also central to its aesthetics . Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". These subjective judgments vary not only from person to person, but also with time and culture: for example, as a proof is obtained, simplified or better understood,

1258-601: A theorem if proven true. Until the end of the 19th century and the foundational crisis of mathematics , all mathematical theories were built from a few basic properties that were considered as self-evident; for example, the facts that every natural number has a successor, and that there is exactly one line that passes through two given distinct points. These basic properties that were considered as absolutely evident were called postulates or axioms ; for example Euclid's postulates . All theorems were proved by using implicitly or explicitly these basic properties, and, because of

1332-556: A theorem of the ambient theory, although they can be proved in a wider theory. An example is Goodstein's theorem , which can be stated in Peano arithmetic , but is proved to be not provable in Peano arithmetic. However, it is provable in some more general theories, such as Zermelo–Fraenkel set theory . Many mathematical theorems are conditional statements, whose proofs deduce conclusions from conditions known as hypotheses or premises . In light of

1406-434: A theorem that was once difficult may become trivial. On the other hand, a deep theorem may be stated simply, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem. Logically , many theorems are of the form of an indicative conditional : If A, then B . Such a theorem does not assert B — only that B

1480-442: A theorem. It comprises tens of thousands of pages in 500 journal articles by some 100 authors. These papers are together believed to give a complete proof, and several ongoing projects hope to shorten and simplify this proof. Another theorem of this type is the four color theorem whose computer generated proof is too long for a human to read. It is among the longest known proofs of a theorem whose statement can be easily understood by

1554-463: A wide range of phenomena. Testing the consequences of a theory is part of the scientific method . Physical theories can be grouped into three categories: mainstream theories , proposed theories and fringe theories . Theoretical physics began at least 2,300 years ago, under the Pre-socratic philosophy , and continued by Plato and Aristotle , whose views held sway for a millennium. During

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1628-450: Is Fermat's Last Theorem , and there are many other examples of simple yet deep theorems in number theory and combinatorics , among other areas. Other theorems have a known proof that cannot easily be written down. The most prominent examples are the four color theorem and the Kepler conjecture . Both of these theorems are only known to be true by reducing them to a computational search that

1702-399: Is a device for turning coffee into theorems" , is probably due to Alfréd Rényi , although it is often attributed to Rényi's colleague Paul Erdős (and Rényi may have been thinking of Erdős), who was famous for the many theorems he produced, the number of his collaborations, and his coffee drinking. The classification of finite simple groups is regarded by some to be the longest proof of

1776-497: Is a model of physical events. It is judged by the extent to which its predictions agree with empirical observations. The quality of a physical theory is also judged on its ability to make new predictions which can be verified by new observations. A physical theory differs from a mathematical theorem in that while both are based on some form of axioms , judgment of mathematical applicability is not based on agreement with any experimental results. A physical theory similarly differs from

1850-419: Is a necessary consequence of A . In this case, A is called the hypothesis of the theorem ("hypothesis" here means something very different from a conjecture ), and B the conclusion of the theorem. The two together (without the proof) are called the proposition or statement of the theorem (e.g. " If A, then B " is the proposition ). Alternatively, A and B can be also termed the antecedent and

1924-403: Is a true statement about a formal system (as opposed to within a formal system) is called a metatheorem . Some important theorems in mathematical logic are: The concept of a formal theorem is fundamentally syntactic, in contrast to the notion of a true proposition, which introduces semantics . Different deductive systems can yield other interpretations, depending on the presumptions of

1998-512: Is generally considered less than 10 to the power 100 (a googol ), there is no hope to find an explicit counterexample by exhaustive search . The word "theory" also exists in mathematics, to denote a body of mathematical axioms, definitions and theorems, as in, for example, group theory (see mathematical theory ). There are also "theorems" in science, particularly physics, and in engineering, but they often have statements and proofs in which physical assumptions and intuition play an important role;

2072-419: Is in contrast to experimental physics , which uses experimental tools to probe these phenomena. The advancement of science generally depends on the interplay between experimental studies and theory . In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations. For example, while developing special relativity , Albert Einstein

2146-443: Is that it is falsifiable , that is, it makes predictions about the natural world that are testable by experiments . Any disagreement between prediction and experiment demonstrates the incorrectness of the scientific theory, or at least limits its accuracy or domain of validity. Mathematical theorems, on the other hand, are purely abstract formal statements: the proof of a theorem cannot involve experiments or other empirical evidence in

2220-636: Is then verified by a computer program. Initially, many mathematicians did not accept this form of proof, but it has become more widely accepted. The mathematician Doron Zeilberger has even gone so far as to claim that these are possibly the only nontrivial results that mathematicians have ever proved. Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identities and hypergeometric identities. Theorems in mathematics and theories in science are fundamentally different in their epistemology . A scientific theory cannot be proved; its key attribute

2294-434: The consequent , respectively. The theorem "If n is an even natural number , then n /2 is a natural number" is a typical example in which the hypothesis is " n is an even natural number", and the conclusion is " n /2 is also a natural number". In order for a theorem to be proved, it must be in principle expressible as a precise, formal statement. However, theorems are usually expressed in natural language rather than in

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2368-532: The EPR thought experiment , simple illustrations of time dilation , and so on. These usually lead to real experiments designed to verify that the conclusion (and therefore the assumptions) of the thought experiments are correct. The EPR thought experiment led to the Bell inequalities , which were then tested to various degrees of rigor , leading to the acceptance of the current formulation of quantum mechanics and probabilism as

2442-557: The Scientific Revolution gathered pace, the concepts of matter , energy, space, time and causality slowly began to acquire the form we know today, and other sciences spun off from the rubric of natural philosophy . Thus began the modern era of theory with the Copernican paradigm shift in astronomy, soon followed by Johannes Kepler 's expressions for planetary orbits, which summarized the meticulous observations of Tycho Brahe ;

2516-442: The collision theory of reactions and energy transfer; unimolecular rate theory and metastable states; condensed-phase and macromolecular aspects of dynamics. Historically, the major field of application of theoretical chemistry has been in the following fields of research: Hence, theoretical chemistry has emerged as a branch of research. With the rise of the density functional theory and other methods like molecular mechanics ,

2590-486: The division algorithm , Euler's formula , and the Banach–Tarski paradox . A theorem and its proof are typically laid out as follows: The end of the proof may be signaled by the letters Q.E.D. ( quod erat demonstrandum ) or by one of the tombstone marks, such as "□" or "∎", meaning "end of proof", introduced by Paul Halmos following their use in magazines to mark the end of an article. The exact style depends on

2664-491: The semantic consequence relation ( ⊨ {\displaystyle \models } ), while others define it to be closed under the syntactic consequence , or derivability relation ( ⊢ {\displaystyle \vdash } ). For a theory to be closed under a derivability relation, it must be associated with a deductive system that specifies how the theorems are derived. The deductive system may be stated explicitly, or it may be clear from

2738-406: The set of all sets cannot be expressed with a well-formed formula. More precisely, if the set of all sets can be expressed with a well-formed formula, this implies that the theory is inconsistent , and every well-formed assertion, as well as its negation, is a theorem. In this context, the validity of a theorem depends only on the correctness of its proof. It is independent from the truth, or even

2812-457: The two-fluid theory of electricity are two cases in this point. However, an exception to all the above is the wave–particle duality , a theory combining aspects of different, opposing models via the Bohr complementarity principle . Physical theories become accepted if they are able to make correct predictions and no (or few) incorrect ones. The theory should have, at least as a secondary objective,

2886-429: The 19th century. Another important event in the 19th century was the discovery of electromagnetic theory , unifying the previously separate phenomena of electricity, magnetism and light. The pillars of modern physics , and perhaps the most revolutionary theories in the history of physics, have been relativity theory and quantum mechanics . Newtonian mechanics was subsumed under special relativity and Newton's gravity

2960-469: The Dutchmen Snell and Huygens. In the 18th and 19th centuries Joseph-Louis Lagrange , Leonhard Euler and William Rowan Hamilton would extend the theory of classical mechanics considerably. They picked up the interactive intertwining of mathematics and physics begun two millennia earlier by Pythagoras. Among the great conceptual achievements of the 19th and 20th centuries were the consolidation of

3034-579: The Earth ) or may be an alternative model that provides answers that are more accurate or that can be more widely applied. In the latter case, a correspondence principle will be required to recover the previously known result . Sometimes though, advances may proceed along different paths. For example, an essentially correct theory may need some conceptual or factual revisions; atomic theory , first postulated millennia ago (by several thinkers in Greece and India ) and

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3108-660: The application of quantum mechanics to problems in chemistry. Other major components include molecular dynamics , statistical thermodynamics and theories of electrolyte solutions , reaction networks , polymerization , catalysis , molecular magnetism and spectroscopy . Modern theoretical chemistry may be roughly divided into the study of chemical structure and the study of chemical dynamics. The former includes studies of: electronic structure, potential energy surfaces, and force fields; vibrational-rotational motion; equilibrium properties of condensed-phase systems and macro-molecules. Chemical dynamics includes: bimolecular kinetics and

3182-419: The applications of relativity to problems in astronomy and cosmology respectively . All of these achievements depended on the theoretical physics as a moving force both to suggest experiments and to consolidate results — often by ingenious application of existing mathematics, or, as in the case of Descartes and Newton (with Leibniz ), by inventing new mathematics. Fourier's studies of heat conduction led to

3256-400: The author or publication. Many publications provide instructions or macros for typesetting in the house style . It is common for a theorem to be preceded by definitions describing the exact meaning of the terms used in the theorem. It is also common for a theorem to be preceded by a number of propositions or lemmas which are then used in the proof. However, lemmas are sometimes embedded in

3330-411: The axioms). The theorems of the theory are the statements that can be derived from the axioms by using the deducing rules. This formalization led to proof theory , which allows proving general theorems about theorems and proofs. In particular, Gödel's incompleteness theorems show that every consistent theory containing the natural numbers has true statements on natural numbers that are not theorems of

3404-514: The body of knowledge of both factual and scientific views and possess a usual scientific quality of the tests of repeatability, consistency with existing well-established science and experimentation. There do exist mainstream theories that are generally accepted theories based solely upon their effects explaining a wide variety of data, although the detection, explanation, and possible composition are subjects of debate. The proposed theories of physics are usually relatively new theories which deal with

3478-407: The context. The closure of the empty set under the relation of logical consequence yields the set that contains just those sentences that are the theorems of the deductive system. In the broad sense in which the term is used within logic, a theorem does not have to be true, since the theory that contains it may be unsound relative to a given semantics, or relative to the standard interpretation of

3552-429: The evidence of these basic properties, a proved theorem was considered as a definitive truth, unless there was an error in the proof. For example, the sum of the interior angles of a triangle equals 180°, and this was considered as an undoubtable fact. One aspect of the foundational crisis of mathematics was the discovery of non-Euclidean geometries that do not lead to any contradiction, although, in such geometries,

3626-458: The first 10 trillion non-trivial zeroes of the zeta function . Although most mathematicians can tolerate supposing that the conjecture and the hypothesis are true, neither of these propositions is considered proved. Such evidence does not constitute proof. For example, the Mertens conjecture is a statement about natural numbers that is now known to be false, but no explicit counterexample (i.e.,

3700-495: The foundations of mathematics to make them more rigorous . In these new foundations, a theorem is a well-formed formula of a mathematical theory that can be proved from the axioms and inference rules of the theory. So, the above theorem on the sum of the angles of a triangle becomes: Under the axioms and inference rules of Euclidean geometry , the sum of the interior angles of a triangle equals 180° . Similarly, Russell's paradox disappears because, in an axiomatized set theory,

3774-416: The highest order, writing Principia Mathematica . In it contained a grand synthesis of the work of Copernicus, Galileo and Kepler; as well as Newton's theories of mechanics and gravitation, which held sway as worldviews until the early 20th century. Simultaneously, progress was also made in optics (in particular colour theory and the ancient science of geometrical optics ), courtesy of Newton, Descartes and

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3848-455: The idea of energy (as well as its global conservation) by the inclusion of heat , electricity and magnetism , and then light . The laws of thermodynamics , and most importantly the introduction of the singular concept of entropy began to provide a macroscopic explanation for the properties of matter. Statistical mechanics (followed by statistical physics and Quantum statistical mechanics ) emerged as an offshoot of thermodynamics late in

3922-412: The inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic . Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as theorems only

3996-484: The interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses. Namely, that the conclusion is true in case the hypotheses are true—without any further assumptions. However, the conditional could also be interpreted differently in certain deductive systems , depending on the meanings assigned to the derivation rules and the conditional symbol (e.g., non-classical logic ). Although theorems can be written in

4070-667: The late 1920s. In the aftermath of World War 2, more progress brought much renewed interest in QFT, which had since the early efforts, stagnated. The same period also saw fresh attacks on the problems of superconductivity and phase transitions, as well as the first applications of QFT in the area of theoretical condensed matter. The 1960s and 70s saw the formulation of the Standard model of particle physics using QFT and progress in condensed matter physics (theoretical foundations of superconductivity and critical phenomena , among others ), in parallel to

4144-448: The most important results, and use the terms lemma , proposition and corollary for less important theorems. In mathematical logic , the concepts of theorems and proofs have been formalized in order to allow mathematical reasoning about them. In this context, statements become well-formed formulas of some formal language . A theory consists of some basis statements called axioms , and some deducing rules (sometimes included in

4218-498: The physical axioms on which such "theorems" are based are themselves falsifiable. A number of different terms for mathematical statements exist; these terms indicate the role statements play in a particular subject. The distinction between different terms is sometimes rather arbitrary, and the usage of some terms has evolved over time. Other terms may also be used for historical or customary reasons, for example: A few well-known theorems have even more idiosyncratic names, for example,

4292-419: The process of becoming established and some proposed theories. It can include speculative sciences. This includes physics fields and physical theories presented in accordance with known evidence, and a body of associated predictions have been made according to that theory. Some fringe theories go on to become a widely accepted part of physics. Other fringe theories end up being disproven. Some fringe theories are

4366-457: The proof of a theorem, either with nested proofs, or with their proofs presented after the proof of the theorem. Corollaries to a theorem are either presented between the theorem and the proof, or directly after the proof. Sometimes, corollaries have proofs of their own that explain why they follow from the theorem. It has been estimated that over a quarter of a million theorems are proved every year. The well-known aphorism , "A mathematician

4440-446: The range of application has been extended to chemical systems which are relevant to other fields of chemistry and physics, including biochemistry , condensed matter physics , nanotechnology or molecular biology . Theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict natural phenomena . This

4514-590: The rise of medieval universities , the only acknowledged intellectual disciplines were the seven liberal arts of the Trivium like grammar , logic , and rhetoric and of the Quadrivium like arithmetic , geometry , music and astronomy . During the Middle Ages and Renaissance , the concept of experimental science, the counterpoint to theory, began with scholars such as Ibn al-Haytham and Francis Bacon . As

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4588-409: The same way such evidence is used to support scientific theories. Nonetheless, there is some degree of empiricism and data collection involved in the discovery of mathematical theorems. By establishing a pattern, sometimes with the use of a powerful computer, mathematicians may have an idea of what to prove, and in some cases even a plan for how to set about doing the proof. It is also possible to find

4662-635: The significance of the axioms. This does not mean that the significance of the axioms is uninteresting, but only that the validity of a theorem is independent from the significance of the axioms. This independence may be useful by allowing the use of results of some area of mathematics in apparently unrelated areas. An important consequence of this way of thinking about mathematics is that it allows defining mathematical theories and theorems as mathematical objects , and to prove theorems about them. Examples are Gödel's incompleteness theorems . In particular, there are well-formed assertions than can be proved to not be

4736-526: The structure of proofs. Some theorems are " trivial ", in the sense that they follow from definitions, axioms, and other theorems in obvious ways and do not contain any surprising insights. Some, on the other hand, may be called "deep", because their proofs may be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself, or show surprising connections between disparate areas of mathematics. A theorem might be simple to state and yet be deep. An excellent example

4810-530: The structures and dynamics of chemical systems and to correlate, understand, and predict their thermodynamic and kinetic properties. In the most general sense, it is explanation of chemical phenomena by methods of theoretical physics . In contrast to theoretical physics, in connection with the high complexity of chemical systems, theoretical chemistry, in addition to approximate mathematical methods, often uses semi-empirical and empirical methods. In recent years, it has consisted primarily of quantum chemistry , i.e.,

4884-484: The study of physics which include scientific approaches, means for determining the validity of models and new types of reasoning used to arrive at the theory. However, some proposed theories include theories that have been around for decades and have eluded methods of discovery and testing. Proposed theories can include fringe theories in the process of becoming established (and, sometimes, gaining wider acceptance). Proposed theories usually have not been tested. In addition to

4958-462: The sum of the angles of a triangle is different from 180°. So, the property "the sum of the angles of a triangle equals 180°" is either true or false, depending whether Euclid's fifth postulate is assumed or denied. Similarly, the use of "evident" basic properties of sets leads to the contradiction of Russell's paradox . This has been resolved by elaborating the rules that are allowed for manipulating sets. This crisis has been resolved by revisiting

5032-472: The theories like those listed below, there are also different interpretations of quantum mechanics , which may or may not be considered different theories since it is debatable whether they yield different predictions for physical experiments, even in principle. For example, AdS/CFT correspondence , Chern–Simons theory , graviton , magnetic monopole , string theory , theory of everything . Fringe theories include any new area of scientific endeavor in

5106-411: The theory (that is they cannot be proved inside the theory). As the axioms are often abstractions of properties of the physical world , theorems may be considered as expressing some truth, but in contrast to the notion of a scientific law , which is experimental , the justification of the truth of a theorem is purely deductive . A conjecture is a tentative proposition that may evolve to become

5180-445: The underlying language. A theory that is inconsistent has all sentences as theorems. The definition of theorems as sentences of a formal language is useful within proof theory , which is a branch of mathematics that studies the structure of formal proofs and the structure of provable formulas. It is also important in model theory , which is concerned with the relationship between formal theories and structures that are able to provide

5254-539: The vision provided by pure mathematical systems can provide clues to how a physical system might be modeled; e.g., the notion, due to Riemann and others, that space itself might be curved. Theoretical problems that need computational investigation are often the concern of computational physics . Theoretical advances may consist in setting aside old, incorrect paradigms (e.g., aether theory of light propagation, caloric theory of heat, burning consisting of evolving phlogiston , or astronomical bodies revolving around

5328-521: The works of these men (alongside Galileo's) can perhaps be considered to constitute the Scientific Revolution. The great push toward the modern concept of explanation started with Galileo , one of the few physicists who was both a consummate theoretician and a great experimentalist . The analytic geometry and mechanics of Descartes were incorporated into the calculus and mechanics of Isaac Newton , another theoretician/experimentalist of

5402-558: Was concerned with the Lorentz transformation which left Maxwell's equations invariant, but was apparently uninterested in the Michelson–Morley experiment on Earth 's drift through a luminiferous aether . Conversely, Einstein was awarded the Nobel Prize for explaining the photoelectric effect , previously an experimental result lacking a theoretical formulation. A physical theory

5476-422: Was given a kinematic explanation by general relativity . Quantum mechanics led to an understanding of blackbody radiation (which indeed, was an original motivation for the theory) and of anomalies in the specific heats of solids — and finally to an understanding of the internal structures of atoms and molecules . Quantum mechanics soon gave way to the formulation of quantum field theory (QFT), begun in

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